Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.8 → 8.3

Time: 19.5s

Precision: 64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.952841730436753784790528219224969402516 \cdot 10^{-166}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right) \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r1026959 = x;
        double r1026960 = y;
        double r1026961 = r1026959 + r1026960;
        double r1026962 = z;
        double r1026963 = t;
        double r1026964 = r1026962 - r1026963;
        double r1026965 = r1026964 * r1026960;
        double r1026966 = a;
        double r1026967 = r1026966 - r1026963;
        double r1026968 = r1026965 / r1026967;
        double r1026969 = r1026961 - r1026968;
        return r1026969;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r1026970 = x;
        double r1026971 = y;
        double r1026972 = r1026970 + r1026971;
        double r1026973 = z;
        double r1026974 = t;
        double r1026975 = r1026973 - r1026974;
        double r1026976 = r1026975 * r1026971;
        double r1026977 = a;
        double r1026978 = r1026977 - r1026974;
        double r1026979 = r1026976 / r1026978;
        double r1026980 = r1026972 - r1026979;
        double r1026981 = -3.952841730436754e-166;
        bool r1026982 = r1026980 <= r1026981;
        double r1026983 = cbrt(r1026975);
        double r1026984 = cbrt(r1026983);
        double r1026985 = r1026984 * r1026984;
        double r1026986 = r1026985 * r1026984;
        double r1026987 = r1026983 * r1026986;
        double r1026988 = cbrt(r1026978);
        double r1026989 = r1026987 / r1026988;
        double r1026990 = r1026983 / r1026988;
        double r1026991 = r1026971 / r1026988;
        double r1026992 = r1026990 * r1026991;
        double r1026993 = r1026989 * r1026992;
        double r1026994 = r1026972 - r1026993;
        double r1026995 = 0.0;
        bool r1026996 = r1026980 <= r1026995;
        double r1026997 = r1026973 * r1026971;
        double r1026998 = r1026997 / r1026974;
        double r1026999 = r1026998 + r1026970;
        double r1027000 = r1026988 * r1026988;
        double r1027001 = cbrt(r1027000);
        double r1027002 = r1026983 / r1027001;
        double r1027003 = cbrt(r1026988);
        double r1027004 = r1026983 / r1027003;
        double r1027005 = r1027004 * r1026992;
        double r1027006 = r1027002 * r1027005;
        double r1027007 = r1026972 - r1027006;
        double r1027008 = r1026996 ? r1026999 : r1027007;
        double r1027009 = r1026982 ? r1026994 : r1027008;
        return r1027009;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.8
Target	8.6
Herbie	8.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -3.952841730436754e-166

   1. Initial program 13.2

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 13.4

      \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]

   4. Applied times-frac 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 7.6

      \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   7. Applied times-frac 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   8. Applied associate-*l* 6.9

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 7.0

       \[\leadsto \left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right) \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   if -3.952841730436754e-166 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

   1. Initial program 50.3

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

   2. Taylor expanded around inf 19.6

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

   if 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 13.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 13.6

      \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]

   4. Applied times-frac 7.9

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 7.9

      \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   7. Applied times-frac 7.9

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   8. Applied associate-*l* 7.3

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 7.3

       \[\leadsto \left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   11. Applied cbrt-prod 7.3

       \[\leadsto \left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\color{blue}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   12. Applied times-frac 7.3

       \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   13. Applied associate-*l* 7.3

       \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.952841730436753784790528219224969402516 \cdot 10^{-166}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right) \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.47542934445772333e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))